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Markus Haase - One of the best experts on this subject based on the ideXlab platform.
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Strip-type Operators and the Logarithm
The Functional Calculus for Sectorial Operators, 2006Co-Authors: Markus HaaseAbstract:As a straightforward abstraction of the logarithm of an injective Sectorial Operator we introduce the notion of a strip-type Operator (Section 4.1). Since the resolvent of a strip-type Operator by definition is bounded outside a horizontal strip, a functional calculus based on Cauchy integrals can be set up (Section 4.2). Section 4.3 is devoted to prove the main result, which states equality between the spectral angle of an injective Sectorial Operator A and the spectral height of the strip-type Operator log A. As a corollary one obtains an important theorem of Pruss and Sohr, saying that in the case where A ∈ BIP, the group type of (A is )s∈ℝ is always larger than the spectral angle of A. In Section 4.4 the problem of ‘inversion’ is discussed, namely the question, which strip-type Operators are actually logarithms of Sectorial Operators. Here we present a theorem of Monniaux, slightly generalised. In Section 4.5 we construct the example of an injective Sectorial Operator A ∈ BIP on a UMD space with the property that the group type of (A is )s∈ℝ is larger than π.
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Perturbation, Interpolation, and Maximal Regularity
Advances in Differential Equations, 2006Co-Authors: Bernhard H. Haak, Markus Haase, Peer Christian KunstmannAbstract:We prove perturbation theorems for Sectoriality and $R$--Sectoriality in Banach spaces, which yield results on perturbation of generators of analytic semigroups and on perturbation of maximal $L^p$--regularity. For a given Sectorial or $R$--Sectorial Operator $A$ in a Banach space $X$ we give conditions on intermediate spaces $Z$ and $W$ such that, for an Operator $S: Z\to W$ of small norm, the perturbed Operator $A+S$ is again Sectorial or $R$--Sectorial, respectively. These conditions are obtained by factorising the perturbation as $S= -BC$, where $B$ acts on an auxiliary Banach space $Y$ and $C$ maps into $Y$. Our results extend previous work on perturbations in the scale of fractional domain spaces associated with $A$ and allow for a greater flexibility in choosing intermediate spaces for the action of perturbation Operators. At the end we illustrate our results with several examples, in particular with an application to a rough boundary value problem.
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SPECTRAL MAPPING THEOREMS FOR HOLOMORPHIC FUNCTIONAL CALCULI
Journal of the London Mathematical Society, 2005Co-Authors: Markus HaaseAbstract:A spectral inclusion theorem and a spectral mapping theorem are proved for the functional calculus for Sectorial Operators. Most proofs are generic, so that similar results can be obtained for other functional calculi. Applications are a new proof for the spectral mapping theorem for fractional powers and the identity for any injective Sectorial Operator.
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Spectral properties of Operator logarithms
Mathematische Zeitschrift, 2003Co-Authors: Markus HaaseAbstract:We prove that the spectral height of the logarithm log A of a Sectorial Operator A equals the spectral angle of A. This yields old results of Pruss/Sohr and McIntosh as corollaries. Then we construct a Sectorial Operator A on a UMD space having bounded imaginary powers such that the group type of (A is ) s ∈ℝ is strictly greater than π.
Lutz Weis - One of the best experts on this subject based on the ideXlab platform.
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Spectral multiplier theorems and averaged R-boundedness
Semigroup Forum, 2017Co-Authors: Christoph Kriegler, Lutz WeisAbstract:Let $A$ be a $0$-Sectorial Operator with a bounded $H^\infty(\Sigma_\sigma)$-calculus for some $\sigma \in (0,\pi),$ e.g. a Laplace type Operator on $L^p(\Omega),\: 1 < p < \infty,$ where $\Omega$ is a manifold or a graph. We show that $A$ has a Hörmander functional calculus if and only if certain Operator families derived from the resolvent $(\lambda - A)^{-1},$ the semigroup $e^{-zA},$ the wave Operators $e^{itA}$ or the imaginary powers $A^{it}$ of $A$ are $R$-bounded in an $L^2$-averaged sense. If $X$ is an $L^p(\Omega)$ space with $1 \leq p < \infty,$ $R$-boundedness reduces to well-known estimates of square sums.
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Maximal Lp-regularity for stochastic evolution equations
2016Co-Authors: Jan Van Neerven, Mark Veraar, Lutz WeisAbstract:Abstract. We prove maximal Lp-regularity for the stochastic evolution equa-tion{ dU(t) +AU(t) dt = F (t, U(t)) dt+B(t, U(t)) dWH(t), t ∈ [0, T], U(0) = u0, under the assumption that A is a Sectorial Operator with a bounded H∞-calculus of angle less than 1 2 pi on a space Lq(O, µ). The driving process WH is a cylindrical Brownian motion in an abstract Hilbert space H. For p ∈ (2,∞) and q ∈ [2,∞) and initial conditions u0 in the real interpolation space DA(1 − 1p, p) we prove existence of unique strong solution with trajectories i
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the h infty functional calculus and square function estimates
arXiv: Functional Analysis, 2014Co-Authors: N J Kalton, Lutz WeisAbstract:Using notions from the geometry of Banach spaces we introduce square functions $\gamma(\Omega,X)$ for functions with values in an arbitrary Banach space $X$. We show that they have very convenient function space properties comparable to the Bochner norm of $L_2(\Omega,H)$ for a Hilbert space $H$. In particular all bounded Operators $T$ on $H$ can be extended to $\gamma(\Omega,X)$ for all Banach spaces $X$. Our main applications are characterizations of the $H^{\infty}$--calculus that extend known results for $L_p$--spaces from \cite{CowlingDoustMcIntoshYagi}. With these square function estimates we show, e. g., that a $c_0$--group of Operators $T_s$ on a Banach space with finite cotype has an $H^{\infty}$--calculus on a strip if and only if $e^{-a|s|}T_s$ is $R$--bounded for some $a > 0$. Similarly, a Sectorial Operator $A$ has an $H^{\infty}$--calculus on a sector if and only if $A$ has $R$--bounded imaginary powers. We also consider vector valued Paley--Littlewood $g$--functions on $UMD$--spaces.
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Perturbation and Interpolation Theorems for the H ^∞-Calculus with Applications to Differential Operators
Mathematische Annalen, 2006Co-Authors: Nigel Kalton, Peer Kunstmann, Lutz WeisAbstract:We prove comparison theorems for the H ^∞-calculus that allow to transfer the property of having a bounded H ^∞-calculus from one Sectorial Operator to another. The basic technical ingredient are suitable square function estimates. These comparison results provide a new approach to perturbation theorems for the H ^∞-calculus in a variety of situations suitable for applications. Our square function estimates also give rise to a new interpolation method, the Rademacher interpolation. We show that a bounded H ^∞-calculus is characterized by interpolation of the domains of fractional powers with respect to Rademacher interpolation. This leads to comparison and perturbation results for Operators defined in interpolation scales such as the L _ p -scale. We apply the results to give new proofs on the H ^∞-calculus for elliptic differential Operators, including Schrödinger Operators and perturbed boundary conditions. As new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded H ^∞-calculus in certain Sobolev spaces and that the Stokes Operator on bounded domains Ω with ∂Ω ∈ C ^1,1 has a bounded H ^∞-calculus in the Helmholtz scale L _ p,σ (Ω), p ∈ (1,∞).
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Real interpolation of domains of Sectorial Operators on Lp-spaces
Journal of Mathematical Analysis and Applications, 2005Co-Authors: Tamara Kucherenko, Lutz WeisAbstract:Abstract Let A be a Sectorial Operator on a non-atomic L p -space, 1 ⩽ p ∞ , whose resolvent consists of integral Operators, or more generally, has a diffuse representation. Then the fractional domain spaces D ( A α ) for α ∈ ( 0 , 1 ) do not coincide with the real interpolation spaces of ( L q , D ( A ) ) . As a consequence, we obtain that no such Operator A has a bounded H ∞ -calculus if p = 1 .
Yuri Tomilov - One of the best experts on this subject based on the ideXlab platform.
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Resolvent representations for functions of Sectorial Operators
Advances in Mathematics, 2017Co-Authors: Charles J. K. Batty, Alexander Gomilko, Yuri TomilovAbstract:Abstract We obtain integral representations for the resolvent of ψ ( A ) , where ψ is a holomorphic function mapping the right half-plane and the right half-axis into themselves, and A is a Sectorial Operator on a Banach space. As a corollary, for a wide class of functions ψ , we show that the Operator − ψ ( A ) generates a Sectorially bounded holomorphic C 0 -semigroup on a Banach space whenever − A does, and the Sectorial angle of A is preserved. When ψ is a Bernstein function, this was recently proved by Gomilko and Tomilov, but the proof here is more direct. Moreover, we prove that such a permanence property for A can be described, at least on Hilbert spaces, in terms of the existence of a bounded H ∞ -calculus for A . As byproducts of our approach, we also obtain new results on functions mapping generators of bounded semigroups into generators of holomorphic semigroups and on subordination for Ritt Operators.
Yu M Arlinskii - One of the best experts on this subject based on the ideXlab platform.
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On m-Sectorial Extensions of Sectorial Operators
'National Academy of Sciences of Ukraine (Co. LTD Ukrinformnauka)', 2017Co-Authors: Yu M Arlinskii, Popov V.a.Abstract:We study maximal Sectorial extensions of an arbitrary closed densely defined Sectorial Operator. In particular, abstract boundary conditions for these extensions are obtained. The results are applied for the parametrization of all m-Sectorial extensions of a nonnegative symmetric Operator in a planar model of two-point interactions
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m-Accretive extensions of a Sectorial Operator
Sbornik: Mathematics, 2013Co-Authors: Yu M Arlinskii, A.b. PopovAbstract:A description of all the maximal accretive extensions and their resolvents is given for a densely defined closed Sectorial Operator in terms of abstract boundary conditions. These results are applied to parametrize all the m-accretive extensions of a symmetric Operator in a planar model of one-centre point interaction. Bibliography: 40 titles.
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Abstract Boundary Conditions for Maximal Sectorial Extensions of Sectorial Operators
Mathematische Nachrichten, 2000Co-Authors: Yu M ArlinskiiAbstract:The purpose of this paper is to give a description of all maximal Sectorial extensions of a given closed densely defined Sectorial Operator with the vertex at the origin in terms of abstract boundary conditions. Applications to ordinary and partial second order differential Operators as well as one-dimensional Schrodinger Operators with a point interaction will be considered.
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On functions connected with Sectorial Operators and their extensions
Integral Equations and Operator Theory, 1999Co-Authors: Yu M ArlinskiiAbstract:For a closed densely defined Sectorial Operator using its Friedrichs and von Neumann-Krein m-Sectorial extensions, the function of two complex variables W _F(λ, z ) which determines the simple part of the Operator up to unitary equivalence, is defined and studied. The strong limit Q _F(λ)=− W _F(λ, −∞) is an analog of the Q-function of positive symmetric Operator. These functions are used for the description of the resolvents of m-Sectorial extensions.
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Closed Sectorial forms and one-parameter contraction semigroups
Mathematical Notes, 1997Co-Authors: Yu M ArlinskiiAbstract:Suppose that s[u, v] is a closed sesquilinear Sectorial form with vertex at zero, half-angle α ∈ [0, π/2), and dense domain D(s) in a Hilbert space H, S is the m -Sectorial Operator associated with s, S _R is the real part of S , and T(t) =exp(− tS ) is the contraction semigroup with generator − S , holomorphic in the sector |arg t |
Charles J. K. Batty - One of the best experts on this subject based on the ideXlab platform.
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Resolvent representations for functions of Sectorial Operators
Advances in Mathematics, 2017Co-Authors: Charles J. K. Batty, Alexander Gomilko, Yuri TomilovAbstract:Abstract We obtain integral representations for the resolvent of ψ ( A ) , where ψ is a holomorphic function mapping the right half-plane and the right half-axis into themselves, and A is a Sectorial Operator on a Banach space. As a corollary, for a wide class of functions ψ , we show that the Operator − ψ ( A ) generates a Sectorially bounded holomorphic C 0 -semigroup on a Banach space whenever − A does, and the Sectorial angle of A is preserved. When ψ is a Bernstein function, this was recently proved by Gomilko and Tomilov, but the proof here is more direct. Moreover, we prove that such a permanence property for A can be described, at least on Hilbert spaces, in terms of the existence of a bounded H ∞ -calculus for A . As byproducts of our approach, we also obtain new results on functions mapping generators of bounded semigroups into generators of holomorphic semigroups and on subordination for Ritt Operators.